symmetry relations should be of thesame variance type, i.e. both upper or both lower.Another important remark is that the symmetry and anti-symmetry characteristic of a ten-sor isinvariantunder coordinate transformations. Hence, a symmetric/anti-symmetrictensor in one coordinate system is symmetric/anti-symmetric in all other coordinate sys-tems. Similarly, a tensor which is neither symmetric nor anti-symmetric in one coordinatesystem remains so in all other coordinate systems.Finally, for a symmetric tensorAijand an anti-symmetric tensorBij(or the other wayaround) we have the following useful and widely used identity:AijBij= 0(97)This is because an exchange of indices will change the sign of one tensor only and this willchange the sign of the term in the summation resulting in having a sum of terms which isidentically zero due to the fact that each term in the sum has its own negation.2.7Exercises2.1 Make a sketch of a rank-2 tensorAijin a 4D space similar to Figure 11. What thistensor looks like?2.2 What are the two main types of notation used for labeling tensors? State two namesfor each.2.3 Make a detailed comparison between the two types of notation in the previous questionstating any advantages or disadvantages in using one of these notations or the other.In this context, like many other contexts in this book, there are certain restrictions on the type andconditions of the coordinate transformations under which such statements are valid. However, thesedetails cannot be discussed here due to the elementary level of this book.
2.7 Exercises74In which context each one of these notations is more appropriate to use than theother?2.4 What is the principle of invariance of tensors and why it is one of the main reasonsfor the use of tensors in science?2.5 What are the two different meanings of the term “covariant” in tensor calculus?2.6 State the type of each one of the following tensors considering the number and positionof indices (i.e. covariant, contravariant, rank, scalar, vector, etc.):aiBjkifbkCji2.7 Define the following technical terms which are related to tensors: term, expression,equality, order, rank, zero tensor, unit tensor, free index, dummy index, covariant,contravariant, and mixed.2.8 Which of the following is a scalar, vector or rank-2 tensor: temperature, stress, crossproduct of two vectors, dot product of two vectors, and rate of strain?2.9 What is the number of entries of a rank-0 tensor in a 2D space and in a 5D space?What is the number of entries of a rank-1 tensor in these spaces?2.10 What is the difference between the order and rank of a tensor considering the differentconventions in this regard?2.11 What is the number of entries of a rank-3 tensor in a 4D space? What is the numberof entries of a rank-4 tensor in a 3D space?
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Tensor, Coordinate system, Polar coordinate system, Coordinate systems
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